Question

If $\int \frac{dx}{32 - 2x^2} = A \log|4 - x| + B \log|4 + x| + c$, then the values of $A$ and $B$ are respectively

• A. $-\frac{1}{8}, \frac{1}{8}$
• B. $\frac{1}{8}, -\frac{1}{8}$
• C. $-\frac{1}{16}, \frac{1}{16}$
• D. $\frac{1}{16}, \frac{1}{16}$

Hint:

You can verify partial fraction constants using differentiation! Take the derivative of the target format: $\frac{d}{dx}[A\log(4-x) + B\log(4+x)] = \frac{-A}{4-x} + \frac{B}{4+x} = \frac{A}{x-4} + \frac{B}{x+4}$. Setting this equal to the initial integrand $\frac{1}{2(4-x)(4+x)}$ lets you determine the matching values for $A$ and $B$ instantly.

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
We are given an indefinite integral problem whose solution format is explicitly matched with logarithmic expansion terms. We need to find the specific constant values of the coefficients $A$ and $B$.

Step 2: Key Formula or Approach:
First, factor out the constants to match the standard special integral template: $$\int \frac{dx}{a^2 - x^2} = \frac{1}{2a} \log\left|\frac{a + x}{a - x}\right| + c = \frac{1}{2a}\log|a + x| - \frac{1}{2a}\log|a - x| + c$$

Step 3: Detailed Explanation:
Let's factor out 2 from the denominator of the given integral: $$I = \int \frac{dx}{2(16 - x^2)} = \frac{1}{2} \int \frac{dx}{4^2 - x^2}$$ Apply the standard inverse quadratic integration rule with $a = 4$: $$I = \frac{1}{2} \cdot \left[ \frac{1}{2(4)} \log\left|\frac{4 + x}{4 - x}\right| \right] + c$$ $$I = \frac{1}{16} \log\left|\frac{4 + x}{4 - x}\right| + c$$ Expand the fraction component using standard logarithmic properties: $$I = \frac{1}{16} \left( \log|4 + x| - \log|4 - x| \right) + c$$ $$I = -\frac{1}{16} \log|4 - x| + \frac{1}{16} \log|4 + x| + c$$ Comparing this directly with the given equation $A \log|4 - x| + B \log|4 + x| + c$: $$A = -\frac{1}{16} \quad \text{and} \quad B = \frac{1}{16}$$ This matches option (C).

Step 4: Final Answer:
The values of $A$ and $B$ are $-\frac{1}{16}$ and $\frac{1}{16}$ respectively, which corresponds to option (C).